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Math 3349, Semester 2 2025 Welcome to Math 3349, Topics in Harmonic Analysis
In this course, we will cover some recent advances in harmonic analysis. I will post a list of topics each week, along with some references, as we go along. Schedule:
Notes by Kevin Zhou References: Terry Tao's Math 254B, Lecture 3 Eli Stein, Harmonic Analysis, Chapter IX, Sections 1-2 Carl Herz, On the mean inversion of Fourier and Hankel transforms, PNAS 1954 Notes by Lekh Bhatia Notes by Dominic Connors References: Section 4.3 of Larry Guth's ICM lecture Terry Tao's Math 254A, Notes for week 2/3 References: Eli Stein, Harmonic Analysis, Chapter IX, Section 4 Andreas Seeger, Chris Sogge, Eli Stein, Regularity properties of Fourier integral operators, Annals Math 1991 Andrew Hassell, Pierre Portal, Jan Rozendaal, Off-singularity bounds and Hardy spaces for Fourier integral operators Trans AMS 2020 Chris Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 1991 Some notes on how local smoothing implies Bochner-Riesz, following Sogge 1991 Some slides I have used in a mini-course at CIRM Notes by Kevin Zhou for week 4 Notes by Noah Gorrell for week 5 Notes by Dominic Connors for week 5 References: Terry Tao's blog post on Stein's spherical maximal theorem Gerd Mockenhaupt, Andreas Seeger, Chris Sogge, Wave Front Sets, Local Smoothing and Bourgain's Circular Maximal Theorem, Annals Math 1992 Notes by Ruiying (Kai) Wu References: Tom Wolff, Recent work connected with the Kakeya problem, Prospects in Mathematics, AMS 1999 (Section 4) Terry Tao's Math 254B, Lectures 1-2 Terry Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke 1999. See paper and his slides here Tony Carbery, Restriction implies Bochner-Riesz for paraboloids, Math Proc Cambridge Philosophical Society 1992 A Tower of Conjectures That Rests Upon a Needle, on Quanta magazine See also this video Tom Wolff, Local smoothing type estimates for large p, GAFA 2000 Isabella Laba and Tom Wolff, A local smoothing estimate in higher dimensions, Journal d'Analyse Mathematique 2002 Terry Tao's Math 247B, Notes 2 Some notes on optimality of ℓp decoupling for the wave equation Notes by Kevin Zhou Notes by Dominic Connors References: Jean Bourgain and Ciprian Demeter, The proof of the l2 decoupling conjecture, Annals Math 2015 Jean Bourgain and Ciprian Demeter, A study guide of the l2 decoupling theorem, Chinese Annals Math 2017 Zane Li, An l2 decoupling interpretation of efficient congruencing: the parabola Revista 2021 Some notes on the Pramanik-Seeger iteration Some slides I have used in a mini-course at CIRM Notes by Ruiying (Kai) Wu References: Jean Bourgain, Ciprian Demeter and Larry Guth, Proof of the main conjecture in Vinogradov’s Mean Value Theorem for degrees higher than three, Annals Math 2016 Trevor Wooley, Nested efficient congruencing and relatives of Vinogradov's mean value theorem, Proc Lond Math Soc 2018 Shaoming Guo, Zane Li, Po-Lam Yung and Pavel Zorin-Kranich, A short proof of l2 decoupling for the moment curve, Amer J Math 2021 Larry Guth, Dominique Maldague and Hong Wang, Improved decoupling for the parabola, J Eur Math Soc 2022 Shaoming Guo, Zane Li and Po-Lam Yung, Improved discrete restriction for the parabola, Math Res Letts 2023 Janos Pach and Micah Sharir, Repeated angles in the plane and related problems, J Comb Theory Series A 1992 Sebastian Herr and Beomjong Kwak, Strichartz estimates and global well-posedness of the cubic NLS on T2, Forum of Math Pi 2024
(See also Section 6 of Terry Tao and Ana Vargas GAFA 2000 for how an argument from [MSS92] deduces wave local smoothing from the Guth-Wang-Zhang square function estimate) Shengwen Gan and Shukun Wu, On local smoothing estimates for wave equations, arXiv 2502.05973 Shengwen Gan, Dominique Maldague and Changkeun Oh, Sharp local smoothing estimates for curve averages, arXiv:2507.09696 Hong Wang and Josh Zahl, The Assouad dimension of Kakeya sets in R3, arXiv 2401.12337, to appear in Invent. Math. Hong Wang and Josh Zahl, Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions, arXiv 2502.17655 Larry Guth, Introduction to the proof of the Kakeya conjecture, arXiv 2505.07695 Larry Guth, Outline of the Wang-Zahl proof of the Kakeya conjecture in R3, arXiv 2508.05475
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